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Published online by Cambridge University Press: 25 January 2013
Let $G$ be a real algebraic semi-simple Lie group and
$\Gamma $ be the fundamental group of a closed negatively curved manifold. In this article we study the limit cone, introduced by Benoist [Propriétés asymptotiques des groupes linéaires. Geom. Funct. Anal. 7(1) (1997), 1–47], and the growth indicator function, introduced by Quint [Divergence exponentielle des sous-groupes discrets en rang supérieur. Comment. Math. Helv. 77 (2002), 503–608], for a class of representations
$\rho : \Gamma \rightarrow G$ admitting an equivariant map from
$\partial \Gamma $ to the Furstenberg boundary of the symmetric space of
$G, $ together with a transversality condition. We then study how these objects vary with the representation.